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2.3 L E S S O N

Rotations

Common Core Math Standards

The student is expected to:

COMMON CORE

G-CO.A.4

Develop definitions of rotations ... in terms of angles, circles, ... and line segments. Also G-CO.A.2, G-CO.A.5, G-CO.B.6

Mathematical Practices

COMMON CORE

MP.5 Using Tools

Language Objective

Students work in small groups or pairs to identify and label the transformation shown on a coordinate plane and if a rotation, identify the point of rotation.

Name

Class

Date

2.3 Rotations

Essential Question: How do you draw the image of a figure under a rotation?

Explore Exploring Rotations

You can use geometry software or an online tool to explore rotations.

A Draw a triangle and label the vertices

A, B, and C. Then draw a point P. Mark P as a center. This will allow you to rotate figures around point P.

Resource Locker

ENGAGE

Essential Question: How do you draw the image of a figure under a rotation?

Possible answer: To draw the image of a figure under a rotation of m? around point P, choose a vertex of the figure, for example, vertex A. Draw? PA. Use a pro_tractor to draw a ray that forms an angle of m? with PA. Use a ruler to mark point A' along the ray so that PA' = PA. Repeat the process with the other vertices of the figure. Connect the images of the vertices (A, B, etc.) to draw the image of the figure. If the figure is on a coordinate plane, use an algebraic rule to find the image of each vertex of the figure. Then connect the images of the vertices.

PREVIEW: LESSON PERFORMANCE TASK

View the online Engage. Discuss the motion of the minute hand of the clock with students. Then preview the Lesson Performance Task.

B Select ABC and rotate it 90? around

point P. Label the image of ABC as ABC . Change the shape, size, or location of ABC and notice how ABC changes.

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C Draw APA, BPB, and CPC . Measure these angles. What do you notice? Does this

relationship remain true as you move point P? What happens if you change the size and shape of ABC?

The measure of each angle is 90?; this remains true regardless of the location of point P or

the size and shape of ABC.

D Measure the distance from A to P and the distance from A' to P. What do you notice? Does

this relationship remain true as you move point P? What happens if you change the size and shape of ABC?

AP = AP'; this remains true regardless of the location of point P or the size and shape

of ABC.

Module 2

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87

Date

Name

2.3 Rotations Exploring Rotations EsCsOCMeOMnROEtNialGGQ-C-uCOO.eA.s.A4t.i2Do,enGv:-eCHloOop.Awd.5ed,foGin-iyCtoiOou.nBsd.6oraf rwottahtieonims ...agienoteframfsigouf arengulensd, ecirrcalerso, t...atiaonnd?line segments. Also You cEanDxAMyupr,oasBaeulwro,gktaaeornPoetrdmraoiasCteanta.rtgeTycleehfsinoegaftnnutewdrrde.arlsTraaeabhwroeioslaruwtaphnnioedllionvpanetlolrliPoitnni.wectetPso.ol to explore rotations.

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Lesson 3

HARDCOVER PAGES 7788

Turn to these pages to

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hardcover student edition.

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87 Lesson 2.3

Reflect

1. What can you conclude about the distance of a point and its image from the center of rotation? A point and its image are both the same distance from the center of rotation.

2. What are the advantages of using geometry software or an online tool rather than tracing paper or a protractor and ruler to investigate rotations? Sample answer: Software or an online tool makes it easy to observe the effect of changing the shape or location of the preimage or changing the location of the center of rotation.

Explain 1 Rotating Figures Using a Ruler and Protractor

A rotation is a transformation around point P, the center of rotation, such that the following is true. ? Every point and its image are the same distance from P. ? All angles with vertex P formed by a point and its image have the same measure. This angle measure

is the angle of rotation.

In the figure, the center of rotation is point P and the angle of rotation is 110?.

Example 1 Draw the image of the triangle after the given rotation.

A

Counterclockwise rotation of 150? around point P

110?

A

C

P

A

B

P

_

_

Step 1 Draw PA. Then use a protractor to draw a ray that forms a 150? angle with PA.

EXPLORE

Exploring Rotations

INTEGRATE TECHNOLOGY

If time permits, students can use the software to experiment with different angles of rotation. In particular, ask students to investigate a 360? angle of rotation. Students should discover that the image of a figure after a 360? rotation coincides exactly with the preimage. Point out that this means a 360? rotation is equivalent to a 0? rotation. Students may also explore angles of rotation greater than 360?. In this case, students should conclude that an equivalent rotation can be found by subtracting 360?(or multiples of 360?) from the angle of rotation.

QUESTIONING STRATEGIES

In what direction does the software rotate figures? How could you use the software to produce a 90? clockwise rotation? Counterclockwise; enter 270? as the angle of rotation.

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C

A

B

P

EXPLAIN 1

Rotating Figures Using a Ruler and Protractor

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PROFESSIONAL DEVELOPMENT

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Learning Progressions

In this lesson, students extend the informal concept of a rotation as a "turn" to a more precise definition. Rotations are one of the three rigid motions that students study in this module (translations and reflections are the other two). Rotations are somewhat more difficult to draw than the other rigid motions and predicting the effect of a rotation may be more difficult for students than predicting the effect of a reflection or a translation. Geometry software is a useful tool for investigating rotations. Students will need a solid understanding of transformations, including rotations, when they combine transformations to solve real-world problems.

INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Encourage students to use their knowledge of

right angles to visualize rotations. Remind students 3/23/14 4:56 AM that a 90? rotation is a quarter turn; a 45? rotation is

half that. For example, suggest students visualize what is approximately a triangle after a rotation of 40? around P.

Rotations 88

QUESTIONING STRATEGIES

How can you use tracing paper to check your construction? Trace the figure; place a pencil's point on P, and rotate the paper counterclockwise for the given angle of rotation. The traced version of the figure should lie on top of the rotated figure.

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Step 2 Use a ruler to mark point A along the ray so that PA = PA. C

A P

A

B

Step 3 Repeat Steps 1 and 2 for points B and C to locate points B and C . Connect points

A, B, and C to draw ABC .

B

C

C

A

P

A

B

B Clockwise rotation of 75? around point Q

_

_

Step 1 Draw QD. Use a protractor to draw a ray forming a clockwise 75? angle with QD.

Step 2 Use a ruler to mark point D along the ray so that QD = QD.

Step 3 Repeat Steps1 and 2 for points E and F to locate points E and F. Connect points D, E, and F to draw DEF.

F E

D

Q

D

E

F Reflect

3. How could you use tracing paper to draw the image of ABC in Part A? Put a piece of tracing paper on the page and trace ABC and point P. Put the point of a pencil on point P and use a protractor to rotate the tracing paper counterclockwise 150?. Trace over ABC in the new location, pressing firmly to make an impression on the page below.

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COLLABORATIVE LEARNING

Small Group Activity GE_MNLESE385795_U1M02L3.indd 89

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Have students work in small groups to write together a description of the similarities and differences they observe among the three transformations: translations, reflections, and rotations. Sample answer: All three transformations preserve the size and shape of the original figure. Each transformation uses a different geometric object (vector, line, or point) to perform the transformation. Translations always preserve the orientation of the original figure, while reflections and rotations may alter the orientation.

Your Turn

Draw the image of the triangle after the given rotation. 4. Counterclockwise rotation of 40? around point P 5. Clockwise rotation of 125? around point Q

J L

K L

K

J

P

U T

T U

S S

Q

Explain 2 Drawing Rotations on a Coordinate Plane

You can rotate a figure by more than 180?. The diagram shows counterclockwise rotations of 120?, 240?, and 300?. Note that a rotation of 360? brings a figure back to its starting location.

y A

When no direction is specified, you can assume that a rotation is counterclockwise. Also, a counterclockwise rotation of x? is the same as a clockwise rotation of (360 - x)?.

The table summarizes rules for rotations on a coordinate plane.

240? 300?

A

Ax 120?

A

Rules for Rotations Around the Origin on a Coordinate Plane

90? rotation counterclockwise

(x, y) (-y, x)

180? rotation

(x, y) (-x, -y)

270? rotation counterclockwise

(x, y) (y, -x)

360? rotation

(x, y) (x, y)

Example 2 Draw the image of the figure under the given rotation.

Quadrilateral ABCD; 270?

The rotation image of (x, y) is (y, -x). Find the coordinates of the vertices of the image.

A(0, 2) A(2, 0) B(1, 4) B'(4, -1) C(4, 2) C'(2, -4) D(3, 1) D'(1, -3)

4 yB

A

-4 -2 0 -2

C Dx 24

-4

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EXPLAIN 2

Drawing Rotations on a Coordinate Plane

QUESTIONING STRATEGIES

How can you predict the quadrant in which the image of the quadrilateral will lie? Every 90? of rotation moves the preimage around the origin by one quadrant, so a 270? rotation moves the preimage from Quadrant I to Quadrant IV.

How can you use the rule for rotation to show that the origin is fixed under the rotation? The rule is (x, y) (y, -x),

so (0, 0) (0, 0), which shows that the origin

is fixed.

AVOID COMMON ERRORS

Some students may confuse the direction of a rotation (clockwise or counterclockwise). Remind students that the direction is assumed to be counterclockwise unless otherwise stated. Associate this default direction with the way the quadrants are numbered.

COMMUNICATING MATH

Students analyze pictures of preimages and images,

and discuss what kind of transformation is shown.

The group must agree before labeling each picture.

If a transformation is identified as a rotation, the

group must determine the point of rotation. Each

picture should be labeled, and this sentence

completed: "This shows a (translation/reflection/

rotation) because

". Provide key terms to

help students complete the statement.

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Rotations 90

91 Lesson 2.3

Predict the quadrant in which the image will lie. Since quadrilateral ABCD lies in Quadrant I and the quadrilateral is rotated counterclockwise by 270?, the image will lie in Quadrant IV.

Plot A, B, C, and D to graph the image.

4 yB

A

C

A D x

-4 -2 0 -2

2 4B

D

-4

C

B KLM; 180?

( ) The rotation image of (x, y) is -x , -y .

Find the coordinates of the vertices of the image.

( ) K(2, -1) K' -2 , 1

M4 y

2

L

K

x

-4 -2 0 K2 4L

( ) L(4, -1) L' -4 , 1

-4 M

( ) M(1, -4) M' -1 , 4

Predict the quadrant in which the image will lie. Since KLM lies in Quadrant IV and

the triangle is rotated by 180?, the image will lie in Quadrant II .

Plot K', L', and M' to graph the image.

Reflect

6. Discussion Suppose you rotate quadrilateral ABCD in Part A by 810?. In which quadrant will the image lie? Explain. Quadrant II; the quadrilateral ABCD is in Quadrant 1. Every rotation of 360? brings the

quadrilateral back to Quadrant I, and since 810? = 360? + 360? + 90?, the 810? rotation is

equivalent to a 90? rotation. This maps the quadrilateral to Quadrant II.

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Lesson 3

LANGUAGE SUPPORT

GE_MNLESE385795_U1M02L3.indd 91

The words rotation and transformation (as well as function and notation) are all

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cognates with Spanish. They contain the same Latin root and have similar

spellings and identical meanings. Point out that all these words in English end

with ?tion, and in Spanish they all end with ?ci?n. This is a word pattern that may

be useful to students who speak English and Spanish.

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